Avoiding a muddy patch in their run

My rabbits' hutch is on the grass with a run which we like to leave attached but which can also be moved around. The ground is quite clay-y, so doesn't drain very well when it's been raining. The grass immediately in front of their hutch, where we stand to feed them and say hello, is gone now and the ground is very compacted, but a bit muddy in rain. I want to replace this area with a harder material, like paving slabs/stepping stones, to avoid having the mud churned up (but keeping the rest of the run grassy). Will that be OK?

My other half is worried that paving slabs will be too hard if the rabbits jump down from the top level (2ft) when we have the door open. Is there something else softer we can use therefore to avoid the muddy patch? Astroturf???!!!

thick layer of bark chippings? So long as they're not poisonous - I know someone on here uses them in their permanent run, with wire mesh underneath to stop digging in/out - so I guess they're OK but it would be nice to know which trees they come from.

Thanks Elve - very good idea. I can use any leftovers on my flower beds too - hopefully it will stop the cats from pooing in them.